J.-J. Levy. Optimal Reductions in the Lambda-Calculus, pages 159--191. Academic Press, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....some redex to be evaluated before we can proceed with the rest, then we can say that we have achieved a flexible system where we have control over what to contract rather than letting reductions force themselves in some order. This may lead to some advantages concerning optimal reductions as in [L evy 80] With our notation, and our new fi reduction, we achieve this flexibility and freedom of choice. Moreover, we do not lose any of the original properties. We have shown in fact that what we provide is a more general fi reduction where more redexes are visible and where all the original ....

L'evy, J.-J. Optimal reductions in the lambda calculus, in To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. Seldin and R. Hindley eds, Academic Press, 1980.

....where we have control over what to contract rather than letting reductions force themselves in some order. Secondly, we think that an investigation concerning the complete class of visible redexes in a term gives a better understanding of reduction strategies, e.g. the optimal reductions as in [L evy 80] 1.2 Why definition mechanisms Practical experiences with type systems show that definitions are indispensable for any realistic application. Without definitions, terms soon become forbiddingly complicated. By using definitions one can avoid such an explosion in complexity. This is, by the ....

L'evy, J.-J. (1980), Optimal reductions in the lambda calculus, in To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. Seldin and R. Hindley eds, Academic Press.

....where we have control over what to contract rather than letting reductions force themselves in some order. Secondly, we think that an investigation concerning the complete class of visible redexes in a term gives a better understanding of reduction strategies, e.g. the optimal reductions as in [L evy 80] 1.2 Why definition mechanisms In many type theories and lambda calculi, there is no possibility to introduce definitions which are abbreviations for large expressions and which can be used several times in a program or 3 a proof. This possibility is essential for practical use, and indeed ....

L'evy, J.-J. Optimal reductions in the lambda calculus, in To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. Seldin and R. Hindley eds, Academic Press, 1980.

....incorporation of full rules makes it necessary to copy environments in body. 5 Conclusion We presented a new formal system called f calculus which stresses sharing as the most essential element and studied the basic properties as a functional calculus. The optimal reduction defined by L evy[7] is closely related to sharing, and two formalisms[3] 8] using environments have been studied from the viewpoint of optimality in L evy s sense. However 3CCL[3] which executes full rules based on categorical combinators is insufficient for optimality. T l RS[8] whose reduction system is weak as ....

J.-J. L'evy. Optimal reductions in the lambda-calculus. To H.B.Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, 1980.

....redexes, i.e. the redexes that do not have residuals under any S normalizing reduction. We show also that S minimal reductions need not exist if S is stable but is not regular. Our study of optimal normalization w.r.t. stable sets S is a generalization of Levy s optimality theory [32], developed for the # calculus. That is, we consider multistep reductions contracting a number of redexes in the same family in parallel, and consider optimality w.r.t. the number of such multisteps. This is chosen because Barendregt et al. [6] showed that no one step optimal recursive # reduction ....

.... case of the # calculus [5] or OCRSs, for co initial reductions P and Q, one can define in OERSs the notion of residual of P under Q, written P Q, using Klop s method of commutative diagrams [29] Klop s method is equivalent to Levy s original definition of the residual relation in the # calculus [31, 32]; the latter is more algebraic in nature and uses multisteps rather than complete developments. We write P Theta Q if P Q = #, where Theta is the Levy embedding relation. P and Q are called Levy equivalent 5 , written P #L Q, if P Theta Q and Q Theta P . It follows from the definition ....

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J.-J. Levy. Optimal reductions in the lambda-calculus. In J. R. Hindley and J. P. Seldin, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda-calculus and Formalism, pages 159--192. Academic Press, 1980.

....coping with implementation there is a lot of theoretical work to be developed. In particular, since our aim is that of providing an optimal implementation, we must start with formalizing the notion of optimality. This means that all the work about Levy s families of redexes in the calculus [Le78, Le80] must be suitably generalized to IS s, and this generalization, as we shall see, is much less evident of what one could imagine. The structure of the paper is the following. We shall start with a detailed discussion about the relations between Interaction Systems, Interaction Nets and Combinatory ....

....Here we have a first surprising result: L evy s zig zag [Le78] is not adequate for IS s (and thus for CRS s) This proves that, despite of its abstract nature, this definition of the family relation is not general enough. Similar problems arise with the generalization of the extraction relation [Le80], that essentially express the causal dependencies of a redex. Only labelling passes quite smoothly to IS s, and this will be defined in section 5. In section 6 we introduce a subclass of IS s: the Discrete Interaction Systems. This class is essentially our correspective of Klop s TRS s. The ....

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J.J.Levy. Optimal reductions in the lambda-calculus. In J.P.Seldin and J.R.Hindley editors, "To H.B.Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism", Academic Press. 1980.

....results has very interesting consequences: if we do not garbage collect any of these memoization nodes (we may since they are weak pointers) then any redex of form can reuse the memoized result, s 1 . This leads to a very practical implementation that approximates optimal lambda reductions [18], with the caveat of using hash consing, of course. The combination of these techniques has proven to be very effective. With hash consing and memoization, common operations such as equality tests, testing if a type is in normal form, and finding out the set of free variables, can all be done in ....

J.-J. Levy. Optimal reductions in the lambda calculus. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Edited by J. P. Seldin and J. R. Hindley, Academic Press, 1980.

....the main theoretical aspects of optimal reductions. In particular, we have defined the notion of redex family via a suitable generalization of Levy s labeling, and we have compared this definition with other well known approaches to the family relation (copy relation, and extraction process [21]) We remark that, up to our knowledge, this has been the first attempt to generalize the theory of optimal reduction to a super system of calculus. The technical preliminaries in [3] that revealed some unexpected problems, especially with the copy relation and the extraction process) was ....

....a labeled expression owning INIT are in a same family if and only if their degrees are the same. This approach to the notion of redex family based on labels does not give much insights about the intuitions that are behind. There are other equivalentapproaches, suggested by the case of calculus [20,21]. The relations among them have been discussed in detail in [3,19] 6 Sharing graphs Let us come to the optimal implementation of IS s. As remarked in the Introduction, the aim is to share, along derivations, redexes that are in the same family. This is yielded by enriching the graphical ....

J.J. Levy. Optimal reductions in the lambda-calculus. In J.P. Seldin and J.R. Hindley, editors, To H.B. Curry, Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 159 -- 191. Academic Press, 1980.

....f = M 0 in let x = y:f y y) in C[y:f y y] Such expressions are indeed copied in lazy functional graph reduction implementations, and we do not view this effect as a shortcoming. Sharing of subterms across different instantiations of bound variables is addressed by optimal reduction strategies (Levy, 1980; Lamping, 1990; Field, 1990; Abadi et al. 1990; Maranget, 1991) Although the additional sharing of those calculi does allow the fewest possible reduction steps, it is not clear how useful optimal reduction is for compilation to efficient low level code. Yoshida (1993) presents a weak lambda ....

Levy, J.-J. (1980). Optimal reductions in the lambda-calculus. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. P. Seldin and J. R. Harding, eds., pages 159--191 Academic Press.

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J.-J. Levy. Optimal Reductions in the Lambda-Calculus, pages 159--191. Academic Press, 1980.

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L'evy, J.-J. Optimal reductions in the lambda calculus, in To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. Seldin and R. Hindley eds, Academic Press, 1980.

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Jean-Jacques Levy. Optimal reductions in the lambda-calculus. In To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, edited by J. P. Seldin and J. R. Hindley, editors), pages 159--191. Academic Press, 1980.

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L'evy, J.-J. Optimal reductions in the lambda calculus, in To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. Seldin and R. Hindley eds, Academic Press, 1980. 23

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Jean-Jacques Levy. Optimal reductions in the lambda-calculus. In To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, edited by J. P. Seldin and J. R. Hindley, editors), pages 159--191. Academic Press, 1980.

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Levy, Jean-Jacques, Optimal Reductions in the LambdaCalculus. In Seldon, J. P. and Hindley, J. R. (editors), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, 1980, pp. 159-191.

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